在群集右設限資料下，Chi和Tsai (2016) 延伸Jeong et al. (2008)於獨立右設限資料下，建立中位數餘命信賴區間的方法至分位數餘命上。學者Ahn和Logan (2016) 利用偽值(pseudo value)於廣義估計方程式 (Generalized Estimating Equation) 來比較多組分位數餘命。由於一般研究者對於偽值不甚了解，所以本論文提出以兩組樣本分位數餘命之差為依據的Wald檢定統計量，來比較兩組分位數餘命。為了建構此統計量，本論文先推導樣本分位數餘命之漸近分布。由於直接估計樣本分位數餘命之變異數的過程較為繁雜，故本論文將採用Tsai et al. (2016)以信賴區間之寬度來間接估計樣本分位數餘命的變異數。此外，本論文也將依據Su和Wei (1993)在獨立右設限資料下，所提出的最小離散檢定統計量，來延伸至群集右設限資料下，建立兩組分位數餘命之差的信賴區間。
經模擬發現，以兩樣本分位數餘命之差為依據的三個Wald檢定統計量，其型一誤差率接近名目型一誤差率，反之，最小離散檢定統計量之型一誤差率會偏低。此外，在存活時間為韋伯分布且兩組群數相等時，三個Wald檢定統計量之檢定力皆較兩個最小離散檢定統計量之檢定力高。綜合兩部分之模擬結果，當存活時間為韋伯分布時，本論文建議研究者使用Wald檢定統計量進行兩組分位數餘命的比較。

To make inference about the difference between two quantile residual lifetimes for clustered survival data, this thesis first extends two minimum dispersion test statistics to clustered survival data. However, the simulation results in Jeong et al. (2008) and Tsai et al. (2016) have shown that the type I error rates of these two minimum dispersion test statistics are too conservative for independent survival data. Therefore, this thesis proposes three Wald type test statistics based on the difference between two sample quantile residual lifetimes with different length-based variance estimators, which are proposed by Tsai et al. (2016). A simulation study is conducted to examine the type I error rates and power of the test statistics proposed in this thesis for clustered survival data.

From the results of simulation study, the type I error rates of the three Wald type test statistics are close to the nominal type I error rate. Whereas, the type I error rates of the two minimum dispersion test statistics are very conservative. Furthermore, we compare the power of the five test statistics under Weibull distribution, and the result shows that the three Wald type test statistics are more powerful than the two minimum dispersion test statistics. Therefore, Wald type test statistics are recommended to be applied in practice when survival times follow Weibull distribution.

[1] Andersen, P. K., Borgan, Ø., Gill, R. D., and Keiding, N. “Statistical Models Based on Counting Processes.” New York: Springer-Verlag. (1993).
[2] Ahn, K. W., and Logan, B. R. “Pseudo-value approach for conditional quantile residual lifetime analysis for clustered survival and competing risks data with applications to bone marrow transplant data.” The Annals of Applied Statistics, 10(2), 618-637. (2016).
[3] Brookmeyer, R., and Crowley, J. “A confidence interval for the median survival time.” Biometrics, 29-41. (1982).
[4] Berger, R. L., Boos, D. D., and Guess, F. M. “Tests and confidence sets for comparing two mean residual life functions.” Biometrics, 103-115. (1988).
[5] Cai, J., and Shen, Y. “Permutation tests for comparing marginal survival functions with clustered failure time data.” Statistics in Medicine, 19(21), 2963-2973. (2000).
[6] Chi, Y., Tsai, T. H., Tu, Y. H., and Tsai, W. Y. “Comparison of several confidence intervals for median residual lifetime with left-truncated and right-censored data.” Communications in Statistics-Simulation and Computation, 45(2), 701-716. (2016).
[7] Chi, Y. and Tsai, T. H. “Two-sample quantile residual life test with clustered survival data.” MOST Technical Report. (2016).
[8] Fleming, Thomas R., and David P. Harrington. “Counting processes and survival analysis.” John Wiley & Sons (1991).
[9] Gangnon, R. E., and Kosorok, M. R. “Sample-size formula for clustered survival data using weighted log-rank statistics.” Biometrika, 263-275. (2004).
[10] Jeong, J. H., Jung, S. H., and Costantino, J. P. “Nonparametric inference on median residual life function.” Biometrics, 64(1), 157-163. (2008).
[11] Kaplan, E. L., and Meier, P. “Nonparametric estimation from incomplete observations.” Journal of the American statistical association, 53(282), 457-481. (1958).
[12] Le, C. T., and Lindgren, B. R. “Duration of ventilating tubes: a test for comparing two clustered samples of censored data.” Biometrics, 328-334. (1996).
[13] Schmittlein, D. C., and Morrison, D. G. “The median residual lifetime: A characterization theorem and an application.” Operations Research, 29(2), 392-399. (1981).
[14] Su, J. Q., and Wei, L. J. “Nonparametric estimation for the difference or ratio of median failure times.” Biometrics, 603-607. (1993).
[15] Tsai, T. H., Tsai, W. Y., Chi, Y., and Chang, S. M. “Confidence intervals for the ratio of two median residual lifetimes with left‐truncated and right‐censored data.” Biometrics. (2016).
[16] Ying, Z., and Wei, L. J. “The Kaplan-Meier estimate for dependent failure time observations.” Journal of Multivariate Analysis, 50(1), 17-29. (1994).
[17] 賴冠瑜，兩組中位數存活時間之比的統計推論，國立成功大學統計學研究所碩士論文。(2014)
[18] 蔡宗憲，左截斷與右設限資料的分位數餘命函數之信賴帶，國立成功大
學統計研究所博士論文。(2015)